Numerical diffusion is a difficulty with computer simulation

Computer simulation

A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

s of continua (such as fluid

Fluid

In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

s) wherein the simulated medium exhibits a higher diffusivity

Diffusivity

Diffusivity can refer to:*Diffusivity of heat*Diffusivity of mass:** Molecular diffusivity ** Eddy diffusivity*Momentum diffusivity...

 than the true medium. This phenomenon can be particularly egregious when the system should not be diffusive at all, for example an ideal fluid acquiring some spurious viscosity in a numerical model.

Explanation

In Eulerian simulations, time and space are divided into a discrete grid and the continuous differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s of motion (such as the Navier–Stokes equation) are discretized

Discretization

In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...

 into finite-difference equations. The discrete equations are in general more diffusive

Diffusion

Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

 than the original differential equations, so that the simulated system behaves differently than the intended physical system. The amount and character of the difference depends on the system being simulated and the type of discretization that is used. Most fluid dynamics or magnetohydrodynamic

Magnetohydrodynamics

Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...

 simulations seek to reduce numerical diffusion to the minimum possible, to achieve high fidelity — but under certain circumstances diffusion is added deliberately into the system to avoid singularities

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

. For example, shock wave

Shock wave

A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

s in fluids and current sheet

Current sheet

A current sheet is an electric current that is confined to a surface, rather than being spread through a volume of space. Current sheets feature in magnetohydrodynamics , the study of the behavior of electrically conductive fluids: if there is an electric current through part of the volume of such...

s in plasma

Plasma (physics)

In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

s are in some approximations infinitely thin; this can cause difficulty for numerical codes. A simple way to avoid the difficulty is to add diffusion that smooths out the shock or current sheet. Higher order numerical methods (including spectral methods) tend to have less numerical diffusion than low order methods.

Example

As an example of numerical diffusion, consider an Eulerian simulation using an explicit time-advance of a drop of green dye diffusing through water. If the water is flowing diagonally through the simulation grid, then it is impossible to move the dye in the exact direction of the flow: at each time step the simulation can at best transfer some dye in each of the vertical and horizontal directions. After a few time steps, the dye will have spread out through the grid due to this sideways transfer. This numerical effect takes the form of an extra high diffusion rate.

When numerical diffusion applies to the components of the momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 vector, it is called numerical viscosity; when it applies to a magnetic field, it is called numerical resistivity

Numerical resistivity

Numerical resistivity is a problem in computer simulations of ideal magnetohydrodynamics . It is a form of numerical diffusion. In near-ideal MHD systems, the magnetic field can diffuse only very slowly through the plasma or fluid of the system; it is rate-limited by the resistivity of the fluid...

.